The effect of lipid composition on membrane curvature originates from the intrinsic shape of different lipid headgroups or acyl chains. Side-by-side packing of lipids will impose curvature on the monolayer according to the shape of the local lipids. Generally, lipids with a small headgroup and large acyl chains, i.e., unsaturated tails, prefer negative curvatures, while lipids with a large headgroup and saturated tails prefer positive curvatures. Owing to the coupling of the lipid bilayer, it was expected that a greater portion of lipids with positive preferred curvature would be found in outer monolayers and those with negative preferred curvature would be found in inner monolayers during vesicle budding. ^{[ 20 ]} This was concurrently proven by the observation that highly curved regions selectively include or exclude specific lipids on the basis of their shape or stiffness and that a significant proportion of lipids with unsaturated tails is found in highly curved membrane tubes. ^{[ 20 – 22 ]} For examples, phosphatidylinositol phosphates (PIs) and lysophosphatidylcholine (LPC) with a large headgroup to acyl tail size ratio show inverted conical shapes and therefore favor intrinsic positive curvatures of the membrane. In contrast, phosphatidylethanolamine (PE), phosphatic acid, diacylglycerol (DAG) and cardiolipin have relatively small headgroups and thus a conical shape, leading to negative membrane curvatures of the monolayer. While the lipid does not have either too large or too small head-to-tail ratio, it is called a cylindrical lipid, for examples, phosphatidylcholine (PC) and phosphatidylserine (PS); these lipids are simply packed to form a flat monolayer. ^{[ 22 – 24 ]}

In addition to headgroup size, saturation of acyl chains also contributes to the headgroup to acyl tail size ratio. Kinks induced by double bonds obviously occupy more space in unsaturated tails than in saturated tails. For example, 1,2-dioleoyl-sn-glycero-3-phosphoserine (DOPS), which was created artificially, favors spontaneous curvature because of its bulky headgroup and monounsaturated acyl chains. ^{[ 25 ]} Polyunsaturated phospholipids (PLs) have been recently reported to increase the ability of dynamin and endophilin to deform and vesiculate synthetic membranes. ^{[ 26 ]}

Imposing the shape of the membrane-binding interface onto the membrane is another effective way of membrane remodeling and is often called as scaffolding (Fig.  1(b) ).

BAR domains are crescent-shaped dimers that bind to membranes and will be the main focus of this review. As a major representative of peripheral membrane-scaffolding proteins, these domains were first noticed as the most conserved feature in amphiphysins from yeast to humans and also in endophilins and nadrins. ^{[ 8 ]} BAR domains can remodel the membrane according to their intrinsic banana shapes, a process that is mediated by electrostatic interactions between positively charged residues (lysine or aginine) on the concave binding surface and negatively charged lipids (PS and/or PI phosphates). ^{[ 27 , 28 ]} BAR domains were previously found to sense membrane curvatures as they bind more readily to liposomes whose curvature is closer to the intrinsic curvature of the BAR. ^{[ 8 , 29 ]} At a higher concentration, this sensor was thought to become an inducer. ^{[ 30 ]} The energetics of amphiphysin on the membrane also show a close relationship between the binding and bending energies during membrane remodeling. ^{[ 31 ]}

An additional feature of the BAR domains is their tight correlation with an N-terminal amphipathic helix (an N-BAR domain). As will be discussed in detail in Section 3, this amphipathic helix also causes membrane bending; many studies have been performed on cooperation between the BAR domain and the amphipathic helix. ^{[ 8 , 32 , 33 ]} Whether the scaffolding mechanism is responsible for the membrane curvature or just serves as a prerequisite for helix insertion is still under debate. In any case, such combination has frequently been noted in membrane remodeling proteins such as amphiphysin, endophilin, BRAP, and nadrin and all these have been reported to induce tubulation in vitro . ^{[ 5 , 34 – 36 ]}

Members of the dynamin family (which are also dimers) were also reported to scaffold inositol lipids, leading to tubulation of the membrane with formation of helical oligomers. ^{[ 29 , 37 ]} The dynamin protein is seldom discussed without mentioning BAR domains as they cooperate with each other to a great extent in many membrane remodeling events such as clathrin-mediated endocytosis. ^{[ 5 , 38 ]} In addition to simple attachment to the membrane, this family of proteins uses GTP hydrolysis to effect membrane fission (scission) which has been discussed in a few early studies. ^{[ 29 , 37 , 39 ]} It is important to acknowledge the role of different proteins that are involved in the scaffolding of membranes; the details of the fission process, however, are beyond the scope of this review.

Another effective way of inducing local membrane curvature is to insert a hydrophobic protein motif between the lipid headgroups. The motif acts like a wedge; when it is inserted near the lipid glycerol backbones, the adjacent hydrocarbon chains tilt and splay to fill the resultant vacant spots underneath the wedge ^{[ 11 ]} (Fig.  1(c) ). However, the lipid response to wedge insertion as local membrane curvatures will dissipate quickly on either side of the insertion, effectiveness of the wedge usually comes with oligomerization of proteins as localization of a number of insertions may cause significantly higher membrane curvatures compared to single insertion. ^{[ 40 , 41 ]}

Sometimes the hydrophobic motif appears on amphipathic helices which are stretches of α -helix with one side being polar (charged) and the other hydrophobic. ^{[ 11 , 30 ]} Amphipathic helices are present in many proteins that participate in different stages of the membrane-remodeling process, such as endophilin, amphiphysin, epsin, nadrin, Arf proteins and the secretion-associated RAS-related 1 (SAR1) GTPase. ^{[ 42 ]} In the case of many of these proteins, the amphipathic helices are unfolded before they insert into membranes. As long as the helices insert into the membranes and fold, they sit flat on the surface with the hydrophobic residues dipping into the hydrophobic phase of the membrane. Another key feature of the amphipathic helices is the nature of their polar face, which grants them various abilities such as sensing membrane curvatures.

Other proteins such as the C2 domains of synaptotagmin-1 and Doc2b also insert hydrophobic loops that are much more bulky than a four-turn helix in order to induce greater curvatures. ^{[ 41 , 43 – 45 ]} The membrane-binding interface of PACSINs, which is a subfamily of the amino-terminal Bin–Amphiphysin–Rvs (F-BAR) domain, and EHDs, which comprise a class of eukaryotic ATPases involved in clathrin-independent endocytosis, also uses hydrophobic loops for insertion. ^{[ 46 , 47 ]} Similar to the amphipathic helices, the nature of these hydrophobic loops varies across the proteins. The membrane-bending activity of C2 domains may be regulated by the Ca ²⁺ concentration. ^{[ 43 , 44 ]}

Transmembrane proteins (commonly known as ion channels, transporters and receptors) that have an intrinsic conical or inverted conical shape can bend associated membranes due to asymmetry of extramembrane domains of the proteins. ^{[ 14 – 18 ]} Protein crowding might also occur for receptors where the lipid bilayer bends from the side with a larger domain. ^{[ 48 ]} Sometimes integral proteins even oligomerize to cause local scaffolding of membranes. ^{[ 49 , 50 ]} Clustering of transmembrane receptors, such as transferrin or low-density lipoprotein (LDL) receptors, was reported to aid in pit stabilization during clathrin-mediated endocytosis. ^{[ 51 ]} Such a crowding mechanism has been suggested to explain the binding of some proteins to the membrane surface at a relatively high concentration but the contribution of crowding to curvatures is still unclear. ^{[ 52 , 53 ]}

Completion of the process of vesicle budding under physiological conditions usually requires a combination of the aforementioned mechanisms. Such a cellular process involves proteins with scaffolding abilities, helix-insertion abilities, or both. Moreover, local curvatures induced by single domains, either by scaffolding by a concave surface or by insertion of a single helix, dissipate rapidly in space; proteins in cells, oligomerize to induce much more pronounced membrane curvatures (Fig.  1(e) ). Further complex cooperation between proteins leads to a large network of interactions that combines sensing, induction, stabilization and even scission of membrane curvatures. ^{[ 12 , 13 , 30 ]}

BAR domain proteins are frequently reported to oligomerize in order to induce macroscopic membrane curvatures such as tubulation and vesiculation. ^{[ 54 , 55 ]} A series of both experimental and computational studies have been conducted for in-depth analysis of BAR oligomers and their contribution to membrane curvatures. ^{[ 56 – 58 ]}

Members of the dynamin family are reported to polymerize into spirals in order to remodel membranes. The fact that they require pre-existing curvatures to assemble efficiently and are GTP-dependent again identifies their main role of inducing scission events instead of early membrane sculpting. ^{[ 59 – 62 ]}

Some coat proteins are formed from oligomerization of single domains such as clathrin, COPI, and COPII, mainly for stabilizing existing membrane curvatures during vesicle budding. ^{[ 63 , 64 ]} Instead of binding directly, they rely on adaptor proteins in order to link to membranes; therefore, this process is sometimes classified as indirect scaffolding. The ability to bend the membranes depends on the rigidity of the assembled polyhedral coat and the transmission of the shape to the membrane. ^{[ 48 ]}

Budding of a clathrin-coated vesicle is a good example of the combination of membrane remodeling mechanisms. Inverted conical lipids (PS and PIP2) asymmetrically distribute between the two membrane monolayers and the intrinsic local curvature is stabilized or even reinforced by proteins with scaffolding or insertion modules; all this is mediated by an oligomeric protein scaffold enclosed by the clathrin coat. ^{[ 13 ]} As oligomerization represents the collective modes of one or more types of proteins, it is extremely complicated and introduces big challenges on both experimental and theoretical fronts.

Shaping of cellular organelles, which also requires global membrane curvatures employs protein oligomerization of caveolin, flotillins, or the reticulons. Similar to clathrin coats, these proteins sometimes utilize bridging proteins to either scaffold or insert motifs into membranes in order to facilitate the formation and stabilization of curvatures at caveolae and the endoplasmic reticulum (ER). ^{[ 65 – 67 ]}

Macroscopic membrane curvatures are present in large intracellular organelles, such as the Golgi or the ER. Scaffolding by actin, intermediate filament (Fig.  1(f) ), and the microtubule cytoskeleton provides such curvatures and also mediates curvatures of plasma membrane in filopodia, lamellipodia, psudopodia, phagocytic cups, and axonal growth cones. The cytoskeleton is therefore also responsible for specific arrangement of the plasma membrane during cell division or phagocytosis. ^{[ 68 , 69 ]} This ability of the cytoskeleton is mainly attributable to membrane tension and involves adjustment of connections to the membrane in which a decrease in tension might help the generation of local curvatures. ^{[ 19 ]} While many endocytosis invagination events were reported to occur together with bursts of actin polymerization, the force-generating role of the cytoskeleton, which has been detected in motility and phagocytosis, is yet to be determined for fission processes. Active membrane pulling and pushing by kinesins, dynein and myosin have been proven to be at least partly responsible for tubulated organelle morphology (e.g., ER, Golgi and endosome) and generation of transport intermediates. ^{[ 70 ]} Surprisingly, the cytoskeleton does not only sculpt membranes on its own but also cooperates with other membrane-bending protein families. Many BAR-domain proteins have been reported to have links by way of signaling proteins to the actin- and microtubule-polymerization machinery, for example, β -centaurins and nadrins. ^{[ 8 ]} Future studies are required on the interplay between the cytoskeleton and membrane-remodeling proteins.

As a member of the BAR superfamily, N-BAR dimerizes to form a concave surface with positively charged residues (Fig.  2(a) ) which was thought to bind electrostatically to the negatively charged moieties on the lipid head groups. This binding event was not observed in computational simulations until Blood et al . ^{[ 91 ]} utilized atomistic MD simulations to show such binding altered the membrane so as to adopt the curvature presented by the concave surface of the N-BAR domain. ^{[ 91 ]} At the same time, orientation of the N-BAR domain was also revealed to be one of the factors affecting its intrinsic curvature toward the membrane surface during the binding process. This study shed light on the role of the N-terminal helix in the N-BAR domain, which was also further studied by atomistic MD simulations. Researchers proposed that the N-terminal helix in the N-BAR domain appears to limit the range of the angles at which the N-BAR domain interacts with the membrane and therefore tends to keep the concave surface facing the membrane. ^{[ 92 ]} This hypothesis was proved by not embedding the amphipathic helices into the bilayer; instead, the helices remained outside the bilayer as random coils throughout the simulations, and it was found N-BAR failed to maintain close association with the bilayer and therefore did not induce any curvature.

One of the aspects that need to be verified for computational studies of membrane sculpting will be whether the simulations could reproduce the membrane curvatures observed in experiments. However, in vivo , membrane curvatures are actually induced by collective motions of membrane remodeling proteins. ^{[ 35 ]} While all-atom MD simulations are too computationally costly to simulate lattices of proteins on the membrane, Arkhipov et al. ^{[ 93 ]} presented coarse-grained (CG) MD simulations of protein lattices on the membrane, producing membrane curvatures comparable to the experimental results. While altering the arrangement of proteins inside the protein lattices could result in a maximum of four-fold effect on the induced membrane curvature, it addressed one of the uncertainties of the CG models, that is, the atomistic information on interactions between proteins inside the lattice was crucial but unclear. Yin et al. ^{[ 56 ]} presented even larger simulations of up to 43 proteins on membranes by using their shape-based coarse-grained (SBCG) models and achieved closing of curved membranes with two ends. A number of speculations have been made on the basis of arrangements of proteins inside the protein lattices producing the membrane curvatures closest to that measured in experiments, but atomic-level evidence was insufficient. CG MD simulations also advanced the understanding of the role of the N-terminal helices in inducing membrane curvatures. Directly removing the amphipathic helices of 8 N-BAR domains in CG MD simulations still generated considerable membrane curvature. ^{[ 57 ]} In addition to the observation that simulations with helices alone fail to recapture any remodeled membrane, Arkhipov et al. ^{[ 57 ]} concluded that scaffolding is the key mechanism by which N-BAR domains bend lipid bilayers. However, the fact that the all-atom MD simulations conducted by Blood et al. ^{[ 92 ]} show membrane curvatures induced solely by amphipathic helices keeps room for debate on the actual role of the N-terminal helix in the N-BAR domain.

A few groups of researchers have studied the membrane-remodeling mechanism of N-BAR domains through simulations of proteins and lipids in water. Ayton et al. ^{[ 94 ]} constructed and refined ^{[ 95 ]} a mesoscopic model which considers the effective curvature field replacing individual proteins to reproduce the membrane remodeling process (Fig.  3(d) ). The model was able to reproduce both tubulation and vesiculation processes with membrane radii comparable to the experiment results. Similar to what revealed by CG MD simulations, parametrization of the model also shed light on the importance of protein density and details of oligomerization. The model was refined to be universal and has been extended to other BAR domains such as F-BAR which will be covered below.

Fig. 3.

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Fig. 3. (a) All-atom MD simulations by Cui et al . [ 97 ] demonstrating symmetrical binding of the N-BAR domain of endophilin protein. Reproduced with permission from Ref. [ 97 ]. Copyright by 2009 Biophysical Society. (b) All-atom MD simulations by Pang et al . [ 80 ] capturing asymmetrical binding of BAR-PH domain protein on PIP 2 -containing membrane. (c) CG MD simulations done by Yu et al . [ 58 ] show the dynamic process of closing of the membrane by an array of F-BAR proteins. Reproduced with permission from Ref. [ 58 ]. Copyright 2013 Yu, Schulten. (d) Mesoscopic model developed by Ayton et al . [ 95 ] in turn succeeded in showing tubulation and vesiculation. Reproduced with permission from Ref. [ 95 ]. Copyright by 2009 Biophysical Society.

The structure of the endophilin N-BAR domain is characterized by two additional amphipathic helices (one on each monomer) located under the top of the concave surface of the N-BAR domain dimer (Fig.  2(b) ). Like N-terminal amphipathic helices, the two additional helices were also highly disordered in solution and were believed to fold upon insertion into the upper membrane leaflet. These additional helices might not only allow endophilin N-BAR domains to induce greater curvatures compared to amphiphysin N-BAR domains and other BAR domain superfamily members, ^{[ 96 ]} but also suggest the possibility of a different binding mechanism compared to that of N-terminal amphipathic helices of amphiphysin. ^{[ 34 ]} The endophilin N-BAR domain has been computationally studied using all-atom MD and CG MD simulations ^{[ 33 , 55 , 97 , 98 ]} concerning the role of the additional inserted helices in the protein-membrane interaction process and their effect on the collective behavior of the proteins, respectively.

Following a similar line of reasoning in the case of the amphiphysin N-BAR domain, although electrostatic interaction between the concave surface of the protein and negatively charged lipids is believed to dominate in membrane curvature induction, the atomistic details of the amphipathic helices, i.e., how they are inserted into the lipid bilayers and how they affect the arrangement of the proteins inside the protein lattice, may indeed be crucial for determining why the resultant membrane curvature induced by members of the N-BAR sub-family differs for different members. Cui et al. ^{[ 97 ]} reported membrane curvatures induced by endophilin N-BAR domains with different orientations of the additional helices using all-atom MD simulations (Fig.  3 ). From the results, a cancellation effect due to the nonparallel arrangement of the additional helices relative to the other two conventional N-terminal amphipathic helices has been revealed. We also note that, on comparing the results of parallel arrangement of the helices and protein in which the additional helices were removed, the induced membrane curvatures were found to be very close. This study indicates that the additional helices might not have contributed much to the curvature induction but might have contributed to some other functional aspects. Later reports ^{[ 55 , 98 ]} suggested that the N-terminal amphipathic helix pairing in the endophilin N-BAR oligomer structure helps to stabilize the lattice. CG MD simulations have shown that the N-BAR system stays much more organized after the simulation compared to the BAR system with deleted N-terminal amphipathic helices. ^{[ 55 , 98 ]} The direct contribution of the additional helices to membrane curvature induction may easily be underestimated and deserves future investigation.

Among the members of the BAR superfamily, members of the F-BAR sub-family (Fig.  2(c) ) are usually good examples of membrane deformation by scaffolding. ^{[ 9 , 54 , 58 , 99 – 101 ]} Nevertheless, comprehensive computational study on lattices of F-BAR domains did not exist until Yu et al . ^{[ 58 ]} published their work (Fig.  3(c) ). They presented the first all-atom MD simulations of F-BAR domains acting on a lipid bilayer in a fully solvated system and also employed an SBCG model to examine the effect of F-BAR domain lattice arrangement on membrane sculpting. The atomistic results revealed that F-BAR domains sculpt membranes according to the scaffolding mechanism. The F-BAR domain does not have unfolded helices like what can be found in N-BAR domains; it acts in three simple and clear steps, namely, binding to the membrane, bending the membrane, and equilibration. Therefore, positively charged residues along the concave surface of the F-BAR domain play a key role in attracting negatively charged membrane lipids, which was also confirmed by mutating the positively charged residue into neutral amino acids. A side loop that can maintain the F-BAR domain in an upright orientation during membrane sculpting can also be captured by all-atom simulations. Membrane tubulation by F-BAR lattices was then simulated using CG approaches, and a range of tube radii comparable to those in experimental results was achieved by constructing different domain arrangements within the lattice. ^{[ 58 ]}

While the membrane-sculpting ability of the BAR domains has been fully recaptured both in vitro ^{[ 82 , 101 , 102 ]} and in cells ^{[ 72 , 103 ]} and is well supported by MD simulations, ^{[ 56 , 58 ]} BAR domains from the protein ACAP1 (Fig.  2(d) ) have recently been found to require an adjacent pleckstrin homology (PH) domain to either bind or bend the membrane. ^{[ 80 ]}

In fact, the PH domain found in GRP1 protein has been previously discussed by Lumb et al . ^{[ 104 ]} with respect to its membrane penetrating ability. All-atom MD simulations have been employed to study the characteristics of several binding pockets of the domain and steered MD (SMD) simulations were used to pull the domain away from the membrane in order to study the energy contribution from the different components of the system to the membrane binding stability. The results have shown that the three putative membrane-binding loops of the PH domain are all able to penetrate the lipid bilayer to a variable degree. Among the three loops in the PH domain, one of them (loop β 6/ β 7: residues 340–353) retained interactions with the bilayer for a longer period relative to the other two loops during the SMD simulations, suggesting that this loop contributes to a more pronounced interaction.

Lai et al. ^{[ 105 ]} used all-atom MD simulations to study the molecular mechanism underlying membrane binding of the GRP1 PH domain. Their results revealed the full picture of the whole binding process of a PH domain with PIP3 lipids in the membrane. Atomic-scale insights highlighted the PS lipids in the membrane, which attract and orient the PH domain in a long range through electrostatic interactions. This interaction reduced the dimensionality of the search process and was believed to increase the residence time of the PH domain on the membrane surface after binding. Longer residence time might confer the PH domain an improved probability of binding to the rare PIP3 lipids that finally stabilize the binding process. However, in contrast to the conclusion drawn by Lumb et al ., ^{[ 104 ]} Lai et al . ^{[ 105 ]} observed the largest number of contacts between the phosphate head of the lipid and the loop β 1/ β 2 (residues 275–283).

Recently, Pang et al. ^{[ 80 ]} reported that an interesting PH domain found in ACAP1 protein adjacent to the BAR domain induced membrane curvatures. While BAR domains are well known for inducing membrane curvature either through scaffolding (represented by F-BAR domains) or helix insertion (represented by N-BAR domains), the BAR-PH domain of ACAP1 seems to rely upon the PH domains for its membrane-binding ability. The BAR-PH dimers were observed to bind the PIP2-containing membrane asymmetrically, i.e., only one of the two identical monomers binds its PH domain to the membrane while the other binds to the adjacent protein. The asymmetrical geometry was not only observed in experiments but was also successfully recaptured in all-atom MD simulations (Fig.  3(d) ). After equilibrium, the BAR domains did not bind to the membrane directly but stayed at a certain distance from the membrane instead. As tubulation was observed with the ACAP1 protein in vitro , this study ^{[ 80 ]} shed light on a new function of the PH domain, which probably took over the conventional role of membrane sculpting of the BAR domains. The BAR domains, in this case, might be involved in interactions between the proteins, thereby affecting the arrangement of the domains inside the protein lattice, which is also crucial for the membrane sculpting.